On almost Cap sets in three variables and the multivariable Cap set problem

TL;DR

This paper establishes bounds on the size of almost cap sets in finite fields and extends the Ellenberg-Gijswijt theorem to multivariable settings, advancing understanding of combinatorial structures in finite vector spaces.

Contribution

It proves size bounds for almost cap sets in _q^n and introduces a multivariable analogue of the Ellenberg-Gijswijt theorem, broadening the scope of cap set research.

Findings

Almost cap sets have size less than c_q^n for some c_q < q.

Established a multivariable analogue of the Ellenberg-Gijswijt theorem.

Provides bounds on subsets avoiding many 3-term arithmetic progressions.

Abstract

In this note we prove that almost cap sets $A \subset \mathbb{F}_q^n$, i.e., the subsets of $\mathbb{F}_q^n$ that do not contain too many arithmetic progressions of length three, satisfy that $|A| < c_q^n$ for some $c_q < q$. As a corollary we prove a multivariable analogue of Ellenberg-Gijswijt theorem.